Hiebert & Carpenter (1992) state that "...school-learned procedures often cannot be used flexibly to solve problems other than those on which they were practiced, and thus, do not transfer well" (p. 79). NetMath is a network project designed by high school upperclassmen, evaluated by high school sophomores, and used by high school freshmen. It is based on the ideals of component software. The components for NetMath were designed and programmed by me with the help of an expert programmer. The components include: a graph, a table, a calculator, a notepad, and digital video. A detailed description of the components appears later in this section. Five high school upper classmen, three seniors and two juniors, will design the World Wide Web (WWW) pages for the project which will integrate the components. In doing so, they will learn an Internet viewer/editor, how to digitize, on-line edit, and integrate the video within Web documents. With the help of the teacher, they will construct the problem sets on rate of change, specifically focusing on linear and non-linear functions. As they are constructing these pages, a sample of high school sophomores will be formative evaluators, viewing the seniorÕs work in progress once a month over the course of a semester. It is hoped that the systematic involvement of the sophomores will help the upperclassmen strengthen their project and will help the upperclassmen become comfortable with showing works in progress. It is also hoped that this evaluation stage will help both the upperclassmen and sophomores becoming more visually literate. The high school freshmen will be the end users of this project. After the freshmen have finished the unit on functions, they will use the problem sets to test their understanding of the concept in terms of real-world applications. All these high school students share the same math teacher, Kathleen Smith, in the Academy at Central High School. When I refer to NetMath, I am referring to the product as a whole. This means, I am not only considering the upperclassmen as designers, the sophomores as evaluators, the freshmen as users, but also that components the make up the Web pages.

In conceptualizing this network project, I hoped to include global participants. I have sent out an email request on various "listservs" and "newsgroups" for outside participation in NetMath. All outside participants would have to have the resources: hardware, software, and expertise, necessary to develop pages for NetMath. The outside participants would have access to the components I developed, but would be responsible for producing and for integrating their own digital video clips for their WWW pages. The sample of sophomores from the Academy at Central High School would be formative evaluators for these pages as well. A sub-study would be conduct to determine: 1) is there a significant difference between the way the sophomores evaluate the pages developed by the upperclassmen in the Academy and those developed by outside participants, and, 2) is there a universal conceptualization of school-based math in terms of real-world applications. However, as of June 4^th, no outside participants have responded to the email request. As such, the project will remained focused on the Academy at Central High School with the hopes that it may be extended globally in the coming months.

Technological Foundations for NetMath

NetMath is modeled after the component-based software, SimCalc (available on-line at: http://tango.mth.umassd.ed/simcalc/), developed by Jeremey Roschelle and Jim Kaput. It also is modeled after Cam Motion developed by TERC researchers (Wanner, Rubin, Osterweil, & Win, see http://hub.terc.edu/terc/view/view_homepage.html for more information). The idea behind component based software like SimCalc stemmed out an alarming realization that even students of advance calculus could not apply basic math concepts to real-world situation (Kaput, 1997, available online at http://tango.mth.umassd.ed/simcalc/simcalquestions.html). The students lacked a deep understanding of the math. To overcome this problem, SimCalc is designed to present multiple and coordinated representations of mathematical concepts. Preliminary research results of the effect of SimCalc on the learning process suggest that giving students an environment where they can explore multiple representations allow the students to form more flexible understandings of complex concepts, like rate of change. For example, in one of the simulated worlds in SimCalc, the students watch two characters, "clown" and "Elvis", walk back and forth on a virtual street. Simultaneously, their movements are tracked on a graph that represents changes in velocity and acceleration. This connection between the simulated environment and the graph better enables the students to understand the relationship between lines on the graph and real-world functions, like walking to meet a friend.

The alternative representations of velocity, distance, and acceleration in both graph and simulated form allow students to develop a better, less simplified understanding of these complex concepts. Moreover, the students are able to interact with the characters or the lines on the graph. This active engagement enhances the learning process as the students explore the relationships between abstract representations like the graph and real-world phenomenon like those simulated. The limitation of SimCalc is that is does not permit digital video as one of the components. Digital video would situate an event and not simply be a representation of the event as are the simulations. NetMath extends SimCalc one step further by including a digital video component. Digital video would situate an actual event unlike the simulations that are representations of an event. Students could shoot the video, digitize it and use it to explain mathematical concepts. For example, instead of watching a clown and Elvis walk up and down the street, there could be actual students walking up and down the street. Moreover, the simulations are limited by technological restraints such as the characters must walk back and forth, not in a continuous motion. Real people do not do this unless they are pacing. Furthermore, including digital video as a component that could also interact with the other components would bring real-world events into the classroom. However, this would require extrapolating points from the video. Currently, there is a piece of software, Cam Motion, which permits this.

Using Cam Motion, students can mark points off on a digital video clip, collect a data set and plot a graph. This innovative use of digital video is unique in its ability to have digital video as an interactive tool. Students are not simply watching the video, but are actively engaged in trying to understand the very nature of the video in mathematical terms. As an independent tool, Cam Motion is strong. Its limitation, however, is the fact that the various attributes in Cam Motion are not completely interrelated with one another. What could make it stronger would be if somehow the interrelatedness of the different attributes could be expressed. Hence, NetMath is an attempt to combine SimCalc and Cam Motion to enhance the learning of mathematical concepts.

NetMath takes advantage of the latest technological innovations and theoretical perspectives to create an environment where dynamic media better represent real-world concepts. Students will solve problem sets relating to real-world issues as part of this network activity. The World Wide Web pages (WWW) supporting this project are designed using the concepts of component-based software and Cam Motion, but those concepts are altered slightly so that they may function in a client/server setting. Functionally, NetMath consists of components programmed through Macromedia Director, an authoring software package. More notably, Macromedia Director allows for programming of interactivity among the components as well as permitting their conversion to Shockwave files, a file format that may be used on the Internet. Similar to SimCalc, the interrelated components include a table, a graph, and a calculator. However, NetMath also has a digital video component similar to Cam Motion. This component was added to the framework because what was lacking in SimCalc was actual associations with the real-world. Although the simulated worlds attempt to represent reality, the students are still removed from the actual context.

Moreover, the design of NetMath reflects some key issues outlined by the National Council for Teachers of Mathematics (NCTM). This network based project encourages learner participation as well as situates school-math concepts within real-world events while utilizing the latest technological innovations. Students will be using technology to construct their learning environment and learning through the technology. The goal of NetMath is to present a universal understanding of the underlying mathematical principles learned in freshmen algebra classes and sophomore geometry classes. Hiebert & Carpenter (1992) have suggested that "the goal of many research and implementation efforts in mathematics education has been to promote learning with understanding" (p.65).

Because the design is similar to SimCalc, the components are all linked to one another. It is this linking that is hoped to encourage understanding and transfer. Additions to one component would update the other components automatically. For example, thinking back to the football problem referenced earlier in this paper, suppose there is a digital video clip of a punter kicking the football. The students by clicking on the video clip could set data points which would follow the progression of the football as it is kicked, in the air, and lands. These data points would then be updated in the table and on the graph while the calculator could be used to determine the appropriate equation. Just with this one problem, the students receive four different perspectives representing the same concept. There is the symbolic notation from the calculator, the organization representation from the table, the visual display of the graph, and the real-life context from the digital video. It is hoped that this type of component software would springboard a student's understanding of what the equations mean in terms of real-world scenarios. Moreover, the fact that each component is automatically updated implies that the students will begin to see the interconnectedness of the relationships. Changing a point or a data set effects the other perspectives. So, for example, if a student wanted to see what would happen if the graph ended before reaching zero, s/he could shorten the curve on the graph which would rewind the video to a certain point where the ball was still mid-air. The students, understanding that gravity would not allow the ball simply to hang in mid-air, could assume a variety of scenarios such as the ball was caught by a defensive player. NetMath attempts to enhance the students' internal representations with social and situated views. Hence, the concepts are taught through their application in the real-world and not in abstraction. Otherwise, students tend to see these abstractions as knowledge itself and do not understand the real-world origins of the concepts; thus, the students do not see the relatedness of their application. NetMath emphasizes the relationships among concepts.

The WWW pages for NetMath will be designed by five high school upperclassmen as part of an independent study. These upperclassmen are working with the same teacher who is the teacher of this study. With the teacher's help, they will determine appropriate problem sets. With my help, they will learn how to digitize video and integrated into a Web page. The upperclassmen will be the primary developers of the network project. The high school freshmen and sophomores will evaluate the project as well as use it as a learning tool.